Work

A simple proof of the 1-dimensional flat chain conjecture

Year: 2025

Type: article

Abstract: We provide a new proof of the fact that metric 1-currents with compact support in the Euclidean space correspond to Federer–Fleming flat chains; that is, the 1-dimensional case of the so-called flat chain conjecture . While previous proofs rely on the delicate task of constructing Lipschitz functions with small L ∞ {L^{\infty}} -norm but large derivative along certain directions at all points of a given Lebesgue null set (the so-called width functions ), our approach is based primarily on Poincaré’s lemma. This perspective allows us to identify a regularity question concerning the solvability of the equation d ⁢ ω = π {d\omega=\pi} for differential k -forms, a question that is closely related to the general validity of the flat chain conjecture in higher dimensions.

Source: Advances in Calculus of Variations

DOI: 10.1515/acv-2025-0025

Authors: Andrea Marchese, Andrea Merlo

Institutions: University of Trento, University of the Basque Country

Language: English

Topic: Geometric Analysis and Curvature Flows

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,712

Funders: Ministero dell’Istruzione, dell’Università e della Ricerca, HORIZON EUROPE European Innovation Council